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This question is similar to that stated in Conformal Welding Reference:

Let $\Sigma$ be a 1-dim compact connected complex manifold with boundary $\partial \Sigma= \partial\Sigma^+ \cup \partial\Sigma^-$, s.t. for every $X \in \pi_0(\partial \Sigma^+)\, (X \in \pi_0(\partial \Sigma^-))$ there is an orientation preserving (reversing) diffeomorphism $\varphi_X: \mathbb{S}^1 \to X$, s.t. $\varphi_X$ extends to a homeomorphism $\hat{\varphi}_{X,r}: \mathbb A_r \to U_{X,r}$ from the half open annuli, bounded by circles with radius $1$ and $r \neq 1$, to an open nbhd $U_{X,r}$ of $X$.

My question is: Which (minimal) additional structure (like complex/real analiticity, etc) should $\hat{\varphi}_{X,r}|_{\text{int}(\mathbb A_r)}$ possess, to make the following possible?

For two such tuples $(\Sigma, \{\varphi_X\}_{X \in \pi_0(\partial \Sigma)})$, $(\Sigma', \{\psi_Y\}_{Y \in \pi_0(\partial \Sigma')})$ I'd like to glue $\Sigma$ and $\Sigma'$ at $X \in \pi_0(\partial \Sigma^+)$ and $Y \in \pi_0(\partial \Sigma'^-)$, such that there exists a unique compatible holomorphic structure on the quotient $\Sigma \amalg \Sigma'/\sim$ with identification coming from $\hat{\varphi}_{X,r}\circ \text{inv}_{\mathbb{C}}\circ\hat{\psi}_{Y,r'}^{-1}$ restricted to $\hat{\psi}_{Y,r'}(\mathbb{A}_{r'} \cap \mathbb{A}_{\frac{1}{r}})$. This should not depend on $r$ and $r'$ up to isomorphism.

EDIT: Question 2: This question is a little more qualitative.. If I drop the extension and assume $\varphi_X$ to be real analytic, then identifying the boundaries should provide a new Riemann surface with unique complex structure, since real analytic maps extend to complex analytic ones uniquely. Why do some people (Neretin, Huang, etc) require extensions of these boundary parametrizations when defining the moduli of Riemann surfaces with parametrized boundaries?

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  • $\begingroup$ In classical conformal welding, a sufficient condition for the existence of conformal welding is that the homeomorphism is quasisymmetric. In that case, the curve obtained is a quasicircle, and you get uniqueness (up to Mobius transformation). The uniqueness follows from the fact that quasicircles are removable for homeomorphisms of the sphere that are conformal on the exterior. In your case, I suppose a sufficient condition to get existence and uniqueness of welding is that the homeomorphisms have a quasiconformal extension. $\endgroup$
    – Malik Younsi
    Commented Jul 9, 2015 at 13:02
  • $\begingroup$ Do you mean that $\hat{\varphi}_{X,r}|_{\text{int}(\mathbb{A}_r)}$ should be quasiconformal? $\endgroup$
    – Bipolar Minds
    Commented Jul 9, 2015 at 14:33
  • $\begingroup$ Yes. I'm not sure I understand Question 2, but usually $\phi_X$ is only assumed to be a homeomorphism on the circle. In that case, even if you assume real analyticity, there might be a problem in the glueing if $\phi_X'$ vanishes somewhere. Maybe this is why some people require that the maps are conformal on a neighborhood of the circle. $\endgroup$
    – Malik Younsi
    Commented Jul 10, 2015 at 15:35
  • $\begingroup$ okay, so requiring $\varphi_X$ to be an analytic diffeomorphism (as it is done by several authors) should be enough to allow a unique gluing operation, right? $\endgroup$
    – Bipolar Minds
    Commented Jul 10, 2015 at 16:05

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